This paper presents an analytical approach to the problem of scalar wave scattering from a slightly rough random surface with the Dirichlet or the Neumann boundary condition. The Green's function method is employed to analyze statistical characteristics of waves including multiple scattering. The integral equations for the first and second moments of scattered waves are derived in a unified form for both boundary conditions. Iterative solutions for the intensity of coherent and incoherent waves are then obtained and are shown to satisfy energy conservation. The effects of multiple scattering due to surface roughness are incorporated systematically into those solutions through an effective surface impedance. Numerical results of the first-order solutions for the incoherent wave show that even for slightly rough surfaces multiple scattering effects are pronounced in the Neumann case at near-grazing incidence when the correlation length of the surface height is smaller than the wavelength. In the Dirichlet case, such effects appear at angles near normal incidence, but are not so significant. The results obtained herein are also compared with those previously obtained under the same perturbation boundary condition.

Analysis of scalar wave scattering from slightly rough random surfaces: A multiple scattering theory

The equations of the correlation functions of irradiance in turbulent media are formally solved for arbitrary order by use of an operator method. A cluster approximation is then applied to the fourth and higher order moments of irradiance to express those in terms of the lower order moments. The values of moments predicted by this approximation show a very good agreement with the recent experimental values observed up to the fourth order moment of irradiance. The irradiance distribution function is then analytically derived from the obtained expression of the moments of irradiance and is found to have the following features: (1) It is close to the Gaussian distribution with respect to the logarithm of irradiance, (2) it has a threshold value for irradiances giving nonvanishing probability, and (3) it has a very small but sharp distribution of the form of the δ function at the threshold value. The spectrum of medium as well as the conditions of the initial wave, e.g., of whether it is a plane wave or is a ...

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Theory of irradiance distribution function in turbulent media

The Dyson equation and the two-frequency Bethe-Salpeter equation for vector-valued electromagnetic waves in the presence of a random distribution of absorptive discrete scatterers with pair correlations are derived on the basis of the Twersky multiple scattering formalism. These equations are subsequently “scalarized” in the case of a tenuous scatterer distribution and within the framework of the Rayleigh-Debye condition. A systematic transition is then made to a two-frequency radiative transfer equation via a phase-space approach. The main strength of the radiative transfer theory expounded here stems from the fact that it is applicable under conditions of large-angle scattering, statistical inhomogeneities and statistical anisotropies. It accounts, also, for a variable scatterer density, absorption, and frequency offsets.

Two‐frequency radiative transfer equation for scalar waves in a random distribution of discrete scatterers with pair correlations

The mutual coherence function of a wave in a random medium with one rough boundary is considered, and the governing equation is obtained in a form of the Bethe-Salpeter (B-S) equation, such that the medium and the boundary are involved on exactly the same footing. The method is based on a previous paper for a purely random boundary, and the present B-S equation is a generalized version of the governing equation when the medium is also random. The equation is compared with the conventional method of solving the transport equation, showing the correspondence of each quantity in detail, and it is concluded that they are equivalent to a sufficient accuracy in the case of one boundary. To obtain the solution, incoherent scattering matrices are introduced independently for both the boundary and the medium, where the former has a direct connection with the conventional scattering cross section per unit area and the latter may be obtained by solving the transport equation. Hence the solution is obtained in several expressions in terms of an effective scattering matrix of the boundary as affected by the random medium and also in terms of a corresponding matrix of the medium free from the boundary. The previous expression by Fung and Eom [ IEEE Trans. Antennas Propag.AP-29, 899 ( 1981)] is reproduced to practical accuracy by the present exact formulation.

Transport theory and boundary-value solutions. I: The Bethe-Salpeter equation and scattering matrices

A systematic theory is given for a general class of scalar waves by introducing a surface Green’s function, which is a 2×2 matrix function governed by boundary equations, transferred onto two reference boundary planes enclosing the real boundary inside. It is subjected to several symmetries, including a relation eventually leading to optical relations. Governing equations of statistical surface Green’s functions of first and second orders are obtained unperturbatively in exactly the same way as for a random medium, on replacing the medium to a surface impedance. Two operator methods are introduced to obtain the surface impedance and integral equations of reflection–transmission coefficients exactly for a given boundary change. The space Green’s functions outside the boundary are obtained by a simple continuation of the surface Green’s functions, and scattering cross sections are obtained from their asymptotic expressions at large distances. Various quantities and equations associated with the incoherent waves are written exactly by the introduction of a scattering matrix, as contrasted with the conventional one for a coherent scatterer. A slightly rough boundary is investigated, with the cross sections for both reflected and transmitted waves, where obtained equations are significant also for higher‐order effects, including multiple scattering. An application to boundary‐value problems in layer transport is suggested.

Statistical theory of scattering and propagation through a rough boundary–The Bethe–Salpeter equation and applications

The random medium is represented by the operator, constructed from the characteristic functional of the medium, and this representation is shown to considerably facilitate the formulation of various equations of waves in random media, as well as obtaining the physical insight into the equations. A specific application is made to waves in the medium of random particles, and the equations obeyed by the characteristic functional of wave are derived with the aid of the effective medium method. Here, the optical condition is exhibited by the condition of an operator in space and time. Independent of this operator method, the general theory is extended, in an unperturbative way, for the equations of the second‐order coherence functions, being given in form of the Bethe–Salpeter equation, and the coherent potential equations are formulated for the basic matrices of two kinds appeared in the equations. The explicit expressions of these matrices are obtained, on utilizing the coherent potential approximation, and are shown to be exactly the same as those obtained by the effective medium method, in both cases of weak‐scattering limit and of random particles. Finally, on employing the appropriate Fourier representations in space and time, the theory is presented in a few different forms, one being particularly suited to derive the equations of multifrequency coherence functions.

K. Furutsu

Papers

Operator methods for time‐dependent waves in random media with applications to the case of random particles